https://www.reddit.com/r/badeconomics/comments/an1f1s/the_fiat_discussion_sticky_come_shoot_the_shit/efr8bc5/

Real energy market Prax hours:

You are head of the commission that designs your countries energy market. The market consists of completely flat demand throughout the year and day and the demand is covered by a nuclear powerplant that produces the exact amount needed. Every market participant has access to a dynamic energy pricing auction. The powerplant is in public hand.

One day, Jill Stein comes along and decides to build a private solar powerplant, which at peak output, supplies the market with 10% of its needs. Financially, this is good for Jill because her solar panels undercut the nuclear powerplant and, at this low relative production, she is able to sell >90% of the power she produced to the same price as nuclear power.

Shitty MS Pint 3D graph

Both are purely fixed cost businesses and the average cost of the solar panels is lower than that of nuclear.

How do you design a price discrimination scheme that prevents this market failure from happening?

I'm posting this as an RI, because this feels like its too big for a comment.

As I stated in the comment chain, this is not a market failure. Jill Stein increases the supply of power with her solar power plant. This causes prices to fall during the day when it is running. If the nuclear power plant must produce excess energy during that period, prices will fall to help reduce the supply of electricity from other sources at that time. Moreover, prices can actually go negative when the supply is too high at certain times; negative electricity prices happen more than you might expect in actual markets. Additionally, prices may even fall so much that nuclear power plants may supply a negative quantity of power rather than stay turned on; this happens when the potential revenue is less than the cost of operation.

That the thing though. Because Solar only works for a few hours a day, that point is never actually reached.

If Jill's output goes to 100% demand during noon, she will sell half of her energy and the powerplant will sell half of it's, but night-time demand is still worth meeting for the nuclear powerplant, only that now it has to charge a little more because of the lost day-time revenues.

There's actually a lot going on here; specifically, the claim that prices will go up because of Jill's decision.

Firstly tl;dr of the model: solar supply goes up, off-peak prices go down.

In order to develop a model, I take an approach similar to Lancaster and develop a model where the consumption of electricity is broken down into two activities: day time consumption and night time consumption (two period model). Additionally, we have two unique firms supplying the electricity and they choose a fixed amount of technology investment that outputs a varying amount of electricity by time. More specifically, solar outputs a different amount of electricity at day time than night, while nuclear outputs a fixed amount at all times. Additionally, just like real life, I assume the electricity market is competitive, so marginal cost equals marginal revenue when firms are maxing profit.

Model

To start, consumers have CD utility from electricity consumption:

[; U = Y_t^{\alpha_t} * Y_s^{\alpha_s} ;]

where [; Y_t ;] is day-time consumption and [; Y_s ;] is night-time consumption. The motivation for CD utility is that there's diminishing returns to having a lot of electricity consumption at any one point; people prefer to balance their electricity consumption through the day. Having close to 0 electricity in any period would make people angery.

Next, we have two producers: (1) a solar plant and (2) a nuclear plant. The production of plant z at time w given by [; \xi\sb{z,w} X_z ;] where [; \xi ;] is the output per unit and [; X ;] is the size of the plant (in arbitrary units). So, for example, the output of the solar plant during the day is given by [; \xi\sb{1t} X_1 ;]. This implies production at each time is given by:

[; Y_t = \xi\sb{1,t} X_1 \xi\sb{2,t} X_2 ;]

and likewise for [; Y_s ;]. Given the budget constraint, [; Y_t p_t Y_s p_s = I ;] where p is the price and I is the budget, the demand curves are [; Y_t = \alpha_t / p_t ;] and [; Y_s = \alpha_s / p_s ;]. Next, suppose we have profit maximizing firms with linear cost. The cost of each technology is c_i and technology is chosen at a fixed quantity for both periods, so total cost for technology 1 is [; C_1 = c_1 X_1 ;] and likewise for technology 2. It is realistic that we can't modify the quantity supplied over the two periods, because people don't uninstall and reinstall solar panels in the middle of the day; additionally, nuclear plants take like 48 hours to turn on and off. Additionally, assuming the cost is linear with the quantity helps keep the math here simple.

Now, I'm going to use this matrix notation to simplify. Addiitionally, let C be a vector of the cost per unit for each technology.

The profit for the two firms is given by: [; \Pi = P^T \, Y - C^T \, X = \left(P^T \, \xi - C^T \right)\, X ;] where [; \Pi ;] is a vector of the profit for each firm. Next, we know [; \xi ;] is invertible. This is true, because we have [; \xi\sb{1,t} > \xi\sb{1,s} ;] and [; \xi\sb{1,t} = \xi\sb{1,s} ;]. That is solar emits more power during the day than night, while we assume nuclear output is constant. In total, this implies our FOC for maximizing profit is this condition.

Now, substituting back into the consumer demand equations, we get the following equilibrium output and input.

Firstly, we know X_i must be positive for both technologies; otherwise, we have an edge case and we can just assume the optimal solution is to invest wholly in one technology. Additionally, we must assume that one technology is more cost effective than the other at one part of the day; otherwise, there would be no reason to adopt the bad one if it was outclassed in both periods. So, assume we have solar being more cost effective during the day and nuclear being more cost effective at night: that is, [; \xi\sb{1,t} / c_1 > \xi\sb{2,t}/c_1 ;] and [; \xi\sb{1,s} / c_1 < \xi\sb{2,s}/c_2 ;]. This plus the condition that X_i is positive implies that we must have [; {\xi\sb{1s}}/{\xi\sb{1t}} < {\xi\sb{2s}}/{\xi\sb{2t}} ;]; this condition states that solar's day-time output to night-time output ratio is higher than for nuclear. Again, we know this is true anyways, since nuclear is constant output and solar works best when the sun is up.

Given these conditions, note that use of X_1 is increasing with its output efficiency. The same applies to X_2. For reference, here's the derivatives of each optimal input wrt their efficiencies and costs. I leave the rest as a proof for the reader.

Secondly, we have the derivatives for the prices. In particular, look at the second element of the derivative of P with respect to [; \xi\sb{1,t};]. I think this is particularly relevant with respect to refuting OP. Firstly, let's suppose solar output during the day was not more cost effective than nuclear. Then, there is no incentive to build a solar plant. However, as solar becomes more cost effective during the day, there does grow an incentive to build a nuclear power plant. Since, solar's cost efficiency increases with its output efficiency, we can think of increases in the output efficiency as increases in solar's viability. That is, it's similar to increasing the quantity of X. The reason we're not looking at X itself is because solving for the price terms as a function of Y and then substituting X while accounting for the general equilibrium effects of increasing X_1 (since optimal X_2 is a function of X_1 as well) is a giant mess. If we want to ignore these general eq effects, then its fairly obvious when substituting [; Y_s = \xi\sb{1,s} X_1 \xi\sb{2,s} X_2 ;] into [; Y_s = \alpha_s / p_s ;] that the derivative with respect to [; X_1 ;] is negative. On the other hand, looking at the derivative of [; p_s ;] with respect to [; \xi\sb{1,t} ;] accommodates gen eq effects while implicitly considering a case with more solar investment.

Anyways, the derivative of [; P_s ;] with respect to [; \xi\sb{1,t} ;] shown at the start of this paragraph is negative; the off-peak price is the second element of vector [; P ;]. Unsurprisingly, as solar becomes more efficient, we will see electricity prices fall during the day as well as during the night. That is, simplifying the equation a bit, we have this. Since we know that nuclear is more cost effective during the night, we have [; \xi\sb{1,s} / c_1 < \xi\sb{2,s}/c_2 \implies \xi\sb{1,s}c_2 < \xi\sb{2,s} c_1 ;] which means that the second term in the numerator is negative. Since the bottom term is obviously positive, prices at night must fall with solar's efficiency during the day.

Numerical Model

Suppose instead for utility, we have this CES function where (1 / (1-phi)) is the intertemporal elasticity of substitution. Although this is continuous, it's discretized easily. I have a numerical model coded with 24 periods using this utility function and everything else staying the same as the model described above. The decision variables are the prices in each period while the objective function is to match supply and demand of electricity in every period. Additionally, solar is technology 2 in this case, while nuclear is the constant output technology 1. The reason supply matches demand even though alpha isn't the same shape as the output of the two technologies is that prices can shift, so demand is actually conforming to the shape of the output.

I run the code with the following parameters:

shift_xi_2= = 1;
parameters.alpha = @(t) 1.75-(2.*t-1).^2/2;
parameters.xi_1  = @(t) 0.*t 1;
parameters.xi_2  = @(t) cos(t.*3-2)/2   shift_xi_2;
parameters.phi = 0.5;
parameters.budget = 1;
parameters.x_1_cost_param = 0.6;
parameters.x_2_cost_param = 1;

Alpha curves upwards during the day, since that's when people use the most electricity. However, the shape of this doesn't really matter than much, since I've coded the rest of it to avoid edge cases. In the following table I shift xi_2 up and down to show how the cost efficiency of solar affects prices at time 0 which is off-peak.

Note that, as solar becomes more efficient and more adopted, the price of electricity goes down in the off-peak as well. This result doesn't change when the cost function is altered to a polynomial or a power function.

Additionally, OP refers to Jill Stein's solar power plant as a market failure. But, this is a literal mistake with the definition of market failure. Moreover, given that a key claim in OPs argument is not true, and the basic rules of supply/demand still apply, his argument as a whole is not applicable. Additionally, utility is also increasing with the efficiency of solar. If consumers are doing better, then obviously this is making things better for people unless you consider producer welfare (best left to scum like corporate shills and Europeans).

edit: Assuming flat demand does not change this result. See here.

Fun Fact

Using the table "Net Generation by State by Type of Producer by Energy Source" from the EIA, I find that nuclear generation was negative three times in the years 2015, 2016, and 2017.

Here's a similar table for fossil fuel generation.